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Linearization of stress rates

Hello all,

 

It would be great if someone is able to provide a step-by-step linearization procedure for objective stress rates (e.g. Oldroyd rate or Jaumann rate) used in finite element implementation of nonlinear material/constitutive laws. I have been looking for this for a long time but it seems thatit is not well documented anywhere in the literature. for example, the Jaumann rate of Kirchhoff stree and it's linearized version in the following paper: Numerical Approximation of Tangent Moduli for Finite Element Implementations of Nonlinear Hyperelastic Material Models; J Biomech Eng. 2008 December; 130(6). I would like to know how to obtain eq. (5) from eq. (3). This is very important step in terms of understanding the finite element implementation of different constittutive theories.

Thank you very much in advance for your help.

Mandar

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